Instability in bacterial populations and the curvature tensor

• Pattern formation in bacterial populations.
• Mobility of bacteria and diffusive processes.
• A decrease of bacterial motility with density can promote separation into bulk phases of two coexisting densities.
• The singularity of the scalar curvature as instability criterion for nonequilibrium systems.

In the geometry associated with equilibrium thermodynamics the scalar curvature RsRs is a measure of the volume of correlation, and therefore the singularities of RsRs indicates the system instabilities. We explore the use of a similar approach to study instabilities in non-equilibrium systems and we choose as a test example, a colony of bacteria. In this regard we follow the proposal made by Obata et al. of using the curvature tensor for studying system instabilities. Bacterial colonies are often found in nature in concentrated biofilms, or other colony types, which can grow into spectacular patterns visible under the microscope. For instance, it is known that a decrease of bacterial motility with density can promote separation into bulk phases of two coexisting densities; this is opposed to the logistic law for birth and death that allows only a single uniform density to be stable. Although this homogeneous configuration is stable in the absence of bacterial interactions, without logistic growth, a density-dependent swim speed v(ρ)v(ρ) leads to phase separation via a spinodal instability. Thus we relate the singularities in the curvature tensor RR to the spinodal instability, that is the appearance of regions of different densities of bacteria.